Forman--Ricci Curvature for Irregular Convex Mosaics

Abstract

Forman has defined a discrete version of the Ricci curvature on Riemannian manifolds, known as the Forman--Ricci curvature. The Forman--Ricci curvature has found significant applications in several pattern recognition problems occurring in natural sciences. Domokos and Langi, on the other hand, have defined a notion of irregularity for convex mosaics, which has also found remarkable applications to the geological problem of fractures in rocks. We define a modification of the classical Forman--Ricci curvature for irregular convex mosaics and demonstrate how they can be used to distinguish between various fractures or cracking patterns appearing in nature.